Journal of Hydrology 365 (2009) 4–10
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Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol
Probabilistic collocation method for unconfined flow in heterogeneous media Liangsheng Shi a,b,*, Jinzhong Yang a, Dongxiao Zhang b,c, Heng Li b a b c
National Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China The Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Kaprielian Hall, 238A 3620 S., Los Angeles, CA 90089, USA Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Article history: Received 19 February 2008 Received in revised form 22 August 2008 Accepted 5 November 2008
Keywords: Probabilistic collocation method Uncertainty Nonstationarity Unconfined flow
s u m m a r y In this study, we make use of the probabilistic collocation method in studying nonlinear flow in heterogeneous unconfined aquifers. In realistic problems there may exist multiple random inputs, nonlinearity, rainfall/evaporation, wells, nonstationary trends and complex boundary conditions. These complications may lead to strong spatial nonstationarity in the flow, which cannot be handled well with most existing approaches. In the present work, a general unconfined problem is explored with the probabilistic collocation method. The stochastic differential equation of flow is reduced into a set of deterministic equations that can be solved with the aid of existing deterministic solvers without any modification. Two types of random inputs are considered, which may be either perfectly correlated or uncorrelated. This approach is demonstrated through some synthetic cases in the presence of rainfall/evaporation, pumping/injection wells, composite mean conductivity fields. The approach can easily combine the proven techniques in deterministic simulations with stochastic treatments and thus provides a practical way for solving stochastic problems with complex fields of hydraulic parameters, nontrivial boundary conditions, and realistic sources/sinks. Published by Elsevier B.V.
Introduction Over the past three decades, many analytical and numerical models have been developed to quantify uncertainty in groundwater and solute transport problems. Recently, collocation methods have been studied and used in different disciplines for uncertainty quantification (e.g. Webster et al., 1996; Tatang et al., 1997; Mathelin et al., 2005; Xiu and Hesthaven, 2005; Ganapathysubramanian and Zabaras, 2006, 2007; Foo et al., 2007; Huang et al.,2007; Li and Zhang, 2007; Ding et al., 2008). On the basis of the techniques for choosing the collocation points, the collocation methods have been classified as probabilistic collocation method (Webster et al., 1996; Tatang et al., 1997), sparse grid collocation method (Ganapathysubramanian and Zabaras, 2006), and Stroud collocation method (Ding et al., 2008). The advantages of the collocation-based methods lie on those deterministic numerical codes can be utilized directly in stochastic modeling. The collocation methods may offer computational advantages over the traditional Monte Carlo method, the moment-equation method (Zhang, 2002), and the stochastic spectral finite element method (SSFEM) (Ghanem and Spanos, 1991). Huang et al. (2007) and Li and Zhang (2007) applied the probabilistic collocation method to the solution of the stochastic differential equations. In their work, the formalism of the collocation method is similar to the SSFEM (Ghanem and Spanos, 1991) in * Corresponding author. Tel.: +1 323 710 3690; fax: +1 213 740 5763. E-mail address:
[email protected] (L. Shi). 0022-1694/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.jhydrol.2008.11.012
the sense that both of them utilize Karhunen–Loeve decomposition and polynomial chaos expansion to represent the input and output random fields. However, the coefficients in the former are computed via a set of independent equations while these coefficients are solved from a set of coupled equations in the latter. In the current work, we make use of the probabilistic collocation method to study the stochastic problem of nonlinear unconfined flow. The probabilistic collocation method is different from the recent work of Xiu and Hesthaven (2005) in that the latter adopts Lagrange polynomials to represent the random outputs and the collocation points are generated by an interpolation method. The problem discussed here can also possibly be solved by other collocation approaches such as the sparse gird collocation and the Stroud collocation method. It is becoming increasingly important to develop an applicable stochastic model that is able to easily incorporate site-specific aquifer conditions. It must posses the flexibility to consider the zonations, layering, recharge (rainfall or evaporation), sinks/ sources, and other complex conditions as most real-world aquifers exhibit (Zhang, 1998; Rubin, 2003; Winter, 2004; Ni and Li, 2006). Unfortunately, most existing methods are too restrictive to handle such practical complexities. The recent development in the collocation methods provides an opportunity to deal with the realistic problems since the collocation methods can reduce the stochastic system into a set of decoupled equations. The decoupled equations can be solved by any existing codes (such as SWMS (Simunek et al., 1992) and Modflow-2000 (Harbaugh et al., 2000)) without
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L. Shi et al. / Journal of Hydrology 365 (2009) 4–10
modifications. These codes’ capability in coping with the complex geologic conditions or boundary conditions can be inherited in the stochastic analysis. Serrano (1995) and Liu et al. (2006) discussed the stochastic nonlinear unconfined flow. Since analytical solution is applied in Serrano’s work (1995), it is not applicable to complex flow. The KLME approach of Liu et al.’s (2006) decomposes the head as a perturbation expansion series in which each term is expanded into a polynomial series of products of orthogonal Gaussian standard random variables. In their work, the resulting equations are reformulated such that they have the same structure of original deterministic governing equation, and the existing flow simulator (Modflow-2000) is used to solve these equations. Since this method needs nontrivial reformulation of the resulting equations, it is hard to be extended to high orders. The present paper is an extension of our earlier work (Li and Zhang, 2007) for confined flow to nonlinear unconfined flow with general inputs and boundary configurations. We examine the application of probabilistic collocation method for unconfined flow in the presence of multiple random inputs, rainfall/evaporation, sinks/sources, and nonstationary conductivity fields. Mathematical formulation Governing equations Let X be a bounded domain in two-dimensional space with boundary oX ¼ CD \ CN , where CD is Dirichlet boundaries and CN is the Neumann boundaries, with CD \ CN ¼ 0. We consider the unconfined flow problem under Dupuit assumption (Bear, 1972)
Sy
ohðx; tÞ ¼ r½KðxÞhðx; tÞrhðx; tÞ þ Rðx; tÞ þ Q ðx; tÞ; ot
x2X
ð1Þ
subject to the following initial and boundary conditions:
hðx; tÞjt¼0 ¼ h0 ðxÞ; x 2 X hðx; tÞ ¼ g D ; x 2 CD
ð2Þ ð3Þ
KðxÞhðx; tÞrhðx; tÞ nðxÞ ¼ g N ;
x 2 CN
ð4Þ
where Sy is specific yield; K(x) is hydraulic conductivity; h(x, t) is the hydraulic head; R(x, t) is areal recharge (e.g. rainfall or evaporation); Q(x, t) is point source/sink term (e.g. pumping or injection well); h0(x) is the initial head in the domain X; gD is the value on the fixed head (Dirichlet) boundaries CD; and gN is the value on the fixed flow (Neumann) boundary CN; n(x) is an outward unit normal of oX. In what follows, we assume Y = ln K and R are second-order stationary random fields with known mean and covariance functions.
Then, we can write the following Karhunen–Loeve expansion of V(x, x):
Vðx; xÞ ¼ hVðxÞi þ
1 pffiffiffiffi X ki fi ðxÞni ðxÞ x 2 X; x 2 X
ð7Þ
i¼1
where ni(x) are orthogonal random variables. Note that Eq. (6) can only be solved analytically for some special covariance functions (i.e. separable exponential covariance function adopted in this work), while in general cases it has to be computed by numerical methods as mentioned by Ghanem and Spanos (1991). In practice, we usually take a truncated KL expansion in the computation. The number of truncated terms depends on the ratio of correlation length to the domain size (Zhang and Lu, 2004). For a fixed flow domain, the smaller the correlation length the large number of terms is to be retained in the KL expansion. Some examples with different correlation lengths will be discussed in the next section. Polynomial chaos expansion Since we do not know the covariance function of outputs before we solve the stochastic differential equation, we utilize an polynomial chaos introduced by Wiener (1938). The original chaos is called homogeneous chaos, and random process is decomposed by Hermite polynomials in terms of Gaussian random variables. In theory, the Hermite polynomial chaos converges to any L2 second-order process with finite variance. However, for non-Gaussian random input variables (e.g. Gamma and uniform), the convergence of Herminte polynomial expansion is not optimal (Xiu and Karniadakis, 2003). Xiu and Karniadakis (2002) proposed generalized polynomial chaos expansions for non-Gaussian distributions. According to different types of random inputs, the polynomials can be chosen from the hypergeometric polynomials of Askey scheme. The general polynomial chaos expansion of the hydraulic head can be written in the form 1 X
Hðx; t; xÞ ¼ a0 ðx; tÞ þ
ai1 ðx; tÞC1 ð1ðxÞÞ
i1 ¼1
þ
i1 1 X X
ai1 i2 ðx; tÞC2 ð1i1 ðxÞ1i2 ðxÞÞ
i1¼1 i2¼1
þ
i1 X i2 1 X X i1¼1 i2¼1 i3¼1
ai1i ðx; tÞC3 ð1i1 ðxÞ1i2 ðxÞ1i3 ðxÞÞ þ 2i3
ð8Þ where Cp ð1i1 ; . . . ; 1ip Þ denotes polynomial of p-order, f1ip g are random variables, fai1 ;...;ip g are coefficients. The polynomials should be chosen properly to assure the best convergence. For example, for Gaussian variables, Gamma, and uniform random variables, Cp ð1i1 ; . . . ; 1ip Þ should be chosen as Hermite, Laguerre, and Legendre polynomials, respectively.
Karhunen–Loeve expansion KL based probabilistic collocation method k;P be a complete probability space, where X is the samLet X; — ple space, P is the probability measure, and k —is the r-algebra of Pmeasurable sets. Consider a random input V(x, x), where x 2 X and x 2 X. The covariance function of this field, CV(x, y) is defined as
C V ðx; yÞ ¼ E½ðVðxÞ hVðxÞiÞðVðyÞ hVðxÞiÞ
ð5Þ
The covariance function is real, symmetric, and positive-definite, thus it has an orthogonal set of eigenfunctions that forms a complete basis (Loeve, 1977). The random input V(x, x) can be represented in terms of infinite orthogonal random variables, eigenvalues ki and eigenfunctions fi(x). The eigenpairs are found by solving the integral equation
Z X
C V ðx; yÞfi ðyÞ dx ¼ ki fi ðxÞ;
x; y 2 X
ð6Þ
In our previous work (Li and Zhang, 2007), the numerical implementation of probabilistic collocation method for confined flow under one random input field has been demonstrated systematically. In this subsection, we present the detailed procedures to solve problem (1) subject to two perfectly correlated or uncorrelated random inputs. By applying the KL expansion, the log-conductivity and recharge can be expressed as
Yðx; xÞ ¼ hYðxÞi þ
MY qffiffiffiffiffi X kYi fiY ðxÞni ðxÞ;
Rðx; t; xÞ
i¼1
¼ hRðx; tÞi þ
MR qffiffiffiffiffi X kRi fiR ðxÞsi ðxÞ i¼1
ð9Þ
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L. Shi et al. / Journal of Hydrology 365 (2009) 4–10
where we have taken a finite-term truncation in Eq. (7); hni nj i ¼ hsi sj i ¼ dij . One has hni sj i ¼ 0 when Y and R are uncorrelated or hni sj i ¼ dij andni ¼ si when Y and R are fully correlated. In the following derivation, vector (n1 ; . . . ; nMY ) and (s1 ; . . . ; sMR ) are denoted as n and s, respectively. The argument (x) will be omitted without causing any confusion. By substituting (9) into governing Eq. (1) yields
Sy
" # " # M Y qffiffiffiffiffi X ohðx; tÞ ¼ r exp hYðxÞi þ kYi fiY ðxÞni hðx; tÞrhðx; tÞ ot i¼1 XMR qffiffiffiffiffiR ð10Þ ki fiR ðxÞsi þ Q ðx; tÞ þ hRðx; tÞi þ i¼1
The random field h(x) is then approximated by a polynomial chaos expansion (we take second-order polynomial chaos for example here), which can be written as
^ tÞ ¼ a ðxÞ þ hðx; o
M X
ai ðx; tÞ1i þ
i¼1
þ
M1 X
M X
i¼1
j¼1
aij ðx; tÞð1i 1j Þ
M X
aii ðx; tÞð12i 1Þ
i¼1
ð11Þ
where M is the dimension of the KL expansions. Eq. (11) is called the M-dimensional polynomial chaos expansion. The basic idea of the probabilistic collocation method is to let the residual of Eq. (10) be zero at particular sets of collocation points. Collocation points are selected by specific algorithm. The algorithm proposed by Webster et al. (1996) is adopted in this work. They construct the collocation points from combinations of the roots of a Hermite polynomial of one order higher than the order of orthogonal polynomial. The selection is also expected to capture regions of high probability (Tatang et al., 1997; Huang et al., 2007; Li and Zhang, 2007). For example, for second-order polynomial chaos expansion with Mp = ffiffiffi2, the pffiffifficollocation points are the combination of three roots ð 3; 0; 3Þ of H3 ð1Þ ¼ 13 31. Since zero has the highest probability for the standard Gaussian random variable with zero mean and unit variance. The first set of collocation points is (0, 0) with the highest probability being captured for each random variable. The other sets of collocation points are selected by keeping as many random variables of high probability as possible. It is noted that not all the combinations of the roots are used in computation. The pffiffiffi p ffiffiffi combinations with lower probability are rejected, such as ð 3; 3Þ. The readers are referred to literature (Webster et al., 1996; Tatang et al., 1997; Huang et al., 2007; Li and Zhang, 2007) for technical details. In this work, since two random inputs are considered, the dimension M of (11) depends on the randomness of Y and R, and the relationship between Y and R. The perfectly correlated and uncorrelated cases are considered. Obviously, when Y and R are fully correlated, they could be expanded by the same series of random variables in their KL expansions. Thus it is feasible to set M ¼ maxfMY ; M R g). And if M = MY, we have 1 ¼ n, otherwise 1 ¼ s. When Y and R are uncorrelated, independent random variables should be assigned in their respective KL expansions. The uncorrelated collocation points are obtained by setting M = MY + MR, and 1 ¼ ðn; sÞ. Therefore, for stochastic problems with multiple random inputs, the independency between each random input leads to a higher dimensionality in the polynomial chaos expansion. For compactness, Eq. (11) can be rewritten as
^ tÞ ¼ hðx;
N X
ci ðx; tÞwi ð1Þ
ð12Þ
i¼1
where there is one-to-one correspondence between the functions Cp ð1i1 ; . . . ; 1ip Þ and wi ð1Þ as well as their corresponding coefficients. N is computed by
N¼
ðp þ MÞ! M!p!
ð13Þ
with p-order polynomial chaos expansion, substituting Eq. (12) into (10), we obtain
" # " # MY qffiffiffiffiffi ^ tÞ X ohðx; Y Y ^ ^ Sy ki fi ðxÞni hðx; tÞrhðx; tÞ ¼ r exp hYðxÞi þ ot i¼1 M R qffiffiffiffiffi X ð14Þ kRi fiR ðxÞsi þ Q ðx; tÞ þ hRðx; tÞi þ i¼1
After substituting the collocation points into (14), it evolves into a deterministic differential equation that has completely the same expression as a deterministic problem. The only difference is that Eq. (14) is computed at selected sets of collocation points. Actually, the probabilistic collocation approach and the Monte Carlo method have the similar working mechanism in the sense that both are implemented by sampling techniques. The Monte Carlo method extracts sample points via direct sampling, Latin hypercube sampling (McKay et al., 1979), or quasi-random sequences (William and Russel, 1994), while PCM chooses the sample points at colloca^ tÞ are computed by standard tion points. In this work, the head hðx; finite element package of SWMS (Simunek et al., 1992). The head field computed from Eq. (14) for a given set of collocation points is called a representation while such a field is referred to as a realization in the Monte Carlo method. This distinction is made because of the different sampling techniques. ^ tÞ are obtained, the coefOnce the representations of head hðx; ficients ci(x, t), i = 1, . . . , N can be solved from
ZCðxÞ ¼ hðxÞ
ð15Þ
where Z, of element zij ¼ wj ð1i Þ, is a space-independent matrix of dimension N N, C(x, t) and h(x, t) are space-dependent and timedependent vectors for each space-time (x, t): Cðx; tÞ ¼ ½c1 ðx; tÞ; ^ 1 ðx; tÞ; h ^2 ðx; tÞ; . . . ; h ^N ðx; tÞ. Then, the c2 ðx; tÞ; . . . ; cN ðx; tÞ; hðx; tÞ ¼ ½h mean and the variance of the hydraulic head can be written as
hhðx; tÞi ¼ c1 ðx; tÞ
r2h ¼
N X
ð16Þ
ci ðx; tÞ2 hw2j i
ð17Þ
i¼2
Numerical examples and comparisons In order to demonstrate the performance of the probabilistic collocation method, we present some synthetic numerical experiments in the presence of rainfall/evaporation, pumping (injection) well, and composite media with different means. The domain of 20 m20 m is uniformly discretized into 41 41 nodes with 3200 triangular elements. In all cases, the boundary C1 (along x = 0, 0 6 y 6 15), C2 (along x = 20, 5 6 y 6 20) are prescribed as constant hydraulic head of 4.5 and 4 m, respectively. The other borders are defined as no-flow boundaries. The boundary conditions are designed to produce a non-unidirectional flow in a bounded domain where the flow is influenced by the boundary conditions in a nontrivial way. For each case, we conduct 10,000 Monte Carlo simulations and compare their results against those from the probabilistic collocation method. In the following testing problems, the random parameters are assumed to be second-stationary and follows a separable exponential covariance function:
jx1 x2 j jy1 y2 j C V ðx; yÞ ¼ r2V exp
g1
g2
ð18Þ
where V could be Y = ln K or R, and g1 and g2 are correlation lengths at x and y directions, respectively. We set the correlation length
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L. Shi et al. / Journal of Hydrology 365 (2009) 4–10
20
length of 2 m and 1 m. Case 4 is designed to have uncorrelated log-conductivity and recharge while case 5 assumes the log-conductivity and recharge to be perfectly correlated. Both cases 4 and 5 have a spatial variable recharge with hRi ¼ 4 cm=day and r2R ¼ 27:49 cm2 =day2 , so that the coefficient of variation for recharge also equals to rR =hRi ¼ 131% . The first 15 eigenpairs are retained to get a finite Karhunen–Loeve expansion of both log-conductivity and recharge for cases 1, 4, and 5 while 30 and 50 eigenpairs are retained for cases 2 and 3, respectively (see Fig. 1). After solving the eigenpairs, the sets of collocation points are then constructed by the algorithm described in the last section. Two sets of collocation points for case 1 are listed in Table 1. The corresponding representations of the conductivity field under these two sets of collocation points are also given in Fig. 2. It is seen that the conductivity fields generated from probabilitic collocation method are typically different than the direct sampling Monte Carlo realizations. The unknowns in equation (15) are finally computed based on the obtained representations. Fig. 3 shows the mean head and head variance of case 1 computed from the Monte Carlo simulation (MCS) and the probabilistic collocation method (PCM). It is seen that the PCM can obtain results comparable to these of the MCS. The mean heads of cases 4 and 5 are quite similar because of the specific settings. Thus, the comparisons between the mean heads from the MCS and PCM in cases 2–5 are not given. Only the head variances along the transection y = 10 m are plotted in Fig. 4. For convenience of estimating the influence of (random) recharge, the variance profile along this transection for case 1, case 1 but with R = 0, cases 4 and 5 are presented in Fig. 3. It is seen the PCM results are close to the Monte Carlo results in all cases. By comparing the case with R = 4 cm/day and that with zero recharge, it is seen that the head variance increase due to the introduction of the recharge and hence the nonlinearity. And the uncorrelated case has larger variance than the fully correlated case. That is to say, the correlation relationship has an impact on the variance distribution.
16
Constant head=4m
No flow
No flow
12 8
Y, m 4
No flow
Constant head=4.5m
0
No flow 0
4
8
North
12
16
20
X, m Figure 1. Sketch of the study domain.
g1 ¼ g2 ¼ 5 m and specific yield Sy = 0.3 in the following cases unless otherwise stated. Cases 1–5. The unconfined flow in the bounded domain subject to constant or random recharge A constant recharge (cases 1–3) or random recharge (cases 4 and 5) is imposed upon the two-dimensional horizontal domain. The introduction of recharge increases the nonlinearity of the flow. Case 1 aims at examining the accuracy of PCM when the flow is affected by a deterministic recharge. The log-conductivity variance r2Y is set as 1.0, and equivalently the the coefficient of variation of conductivity equals to rK =hKi ¼ 131%. A positive constant recharge (e.g. rainfall) R = 4 cm/day is given. Cases 2 and 3 have the same configuration as case 1 except for the respective correlation Table 1 The 5th and 50th set of collocation points. n2
n3
n4
n5
n6
n7
n8
n9
n10
n11
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 pffiffiffi 3
0 0
0 0
20 10.4 9.8 9.2 8.6 8 7.4 6.8 6.2 5.6 5
15
10
5
0
n12 pffiffiffi 3 0
b 20
10
5
0
5
10
x/m
15
20
0
n13
n14
n15
0 0
0 pffiffiffi 3
0 0
14.5 13.5 12.5 11.5 10.5 9.5 8.5 7.5 6.5 5.5 4.5 3.5
15
y/m
a
y/m
5th 50th
n1
0
5
10
15
20
x/m
Figure 2. Representations of conductivity field for the (a) 5th set and (b) 50th set of collocation points (case 1).
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L. Shi et al. / Journal of Hydrology 365 (2009) 4–10
a
b
20
15
y, m
y, m
15
10
5
0
20
10
5
0
5
10
15
20
x, m
0
0
2
4
6
8
10 12 14 16 18 20
x, m
Figure 3. Comparison of mean head (a) and head variance (b) between PCM and MCS (case 1). The solid lines stand for the results from Monte Carlo simulation, and the dash lines from PCM.
The variance profiles of cases 2 and 3 along transection y = 10 m are given in Fig. 5. It is seen the PCM performs well in the case of a smaller correlation scale and thus a larger number of correlation scales within the domain (compared to case 1). When there is a larger number of correlation scales in the domain (as in cases 2 and 3), the influence of boundary conditions is less. This result indicates that the good agreement found in case 1 could not attribute to the influence of the deterministic boundary conditions. The total number of simulations in the PCM for case 1, case 2, and case 3 is 231, 496, and 1326, respectively while it is found that approximately 3000, 4000, and 5000 realizations are needed for Monte Carlo simulation to get the convergent solutions, respectively. The computational cost of case 5 is the same as in case 1 because the same sets of collocation points are used to represent the log-conductivity and recharge fields. However, for case 4, uncorrelated sets of collocation points are generated because of uncorrelated Y and R. In case 4, a 30-term KL expansion is used, with the first 15 collocation points for representing the log-conductivity and the latter 15 for the recharge. Thus, the total number of equations is (30 + 2)!/(30!2!), which is still less than the necessary number of realizations in the Monte Carlo simulation. The Monte Carlo simulation needs about 4000 realizations in case 4. Case 6. The unconfined flow in the presence of pumping well
rate of 20 m3/day is placed at the center of the domain. As shown in Fig. 6, the pumping well leads to a rapid drop of groundwater level near the well. A drastic increase of the head variance is observed near the well. The results from the PCM agree quite well with those from the Monte Carlo simulation. Case 7. The unconfined flow with nonstationary trends Based on case 1, a nonstationary mean is considered in this case. As done in Zhang (1998) and Ni and Li (2005), the hydraulic conductivity is assumed to exhibit both a randomly varying smallscale fluctuation and a large-scale nonstationarity. The meanmoved fluctuation is assumed to be a second-order stationary field with a given covariance functions. The large-scale nonstationarity is represented as a deterministic trend (either continuous or discontinuous) (Ni and Li, 2005). Fig. 7 shows the mean log-conductivity field for case 7. The mean log-conductivity has four zones with hY 1 i ¼ 0 (2 6 x 6 6, 2 6 y 6 6) , hY 2 i ¼ 1 (8 6 x 6 12, 3 6 y 6 10), hY 3 i ¼ 2:5 (9 6 x 6 15, 12 6 y 6 17), and hY 4 i ¼ 2 (the rest of the domain). The flow exhibits strong nonuniformity because of the overall nonstationarity of the conductivity. It is seen from Fig. 7 that despite the strong trending nonstationarity, the PCM gives accurate mean and variance fields at a much smaller computational cost compared to the MCS. Case 8. The unconfined flow with complex configuration
Base on case 1, a pumping well is added to examine the ability of the PCM to treat the situations with a rapidly varying head over a small range near the well. All the parameters remain the same as case 1 except that a pumping well extracting groundwater at the
In this case, we try to simulate a more realistic groundwater flow problem. It is designed to test the robustness of the probabi-
Figure 4. Comparison of head variance along profile y = 10 m between PCM and MCS.
Figure 5. Comparison of head variance along profile y = 10 m between PCM and MCS (cases 2 and 3).
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a
b
20
15
y, m
y, m
15
10
5
0
20
10
5
0
5
10
15
0
20
0
5
x, m
10
15
20
x, m
Figure 6. Comparison of mean head (a) and head variance (b) between PCM and MCS (case 6). The solid lines stand for the results from Monte Carlo simulation, and the dash lines from PCM.
a
b
20
15
y, m
y, m
15
10
10
5
5
0
20
0
5
10
15
0
20
0
5
x, m
10
15
20
x, m
Figure 7. Comparison of mean head (a) and head variance (b) between PCM and MCS (case 7). The solid lines stand for the results from Monte Carlo simulation, and the dash lines from PCM.
a
b
20
15
y, m
y, m
15
10
5
0
20
10
5
0
5
10
x, m
15
20
0
0
5
10
15
20
x, m
Figure 8. Comparison of mean head (a) and head variance (b) between PCM and MCS (case 8). The solid lines stand for the results from Monte Carlo simulation, and the dash lines from PCM.
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listic collocation method. There is strong evaporation (negative recharge) over the whole region. An injection well is placed at the center of the domain with an injection flux 20 m3/day. The conductivity has the same distribution as in case 7. The statistics of the 2 evaporation is set as hRi ¼ 4 cm=day and r2R ¼ 27:49 cm2 =day . The head field exhibits obvious nonstationarity because of the interaction of the bounded domain, the boundary conditions, the regional evaporation, the injection well, and the distributed mean conductivity. It is seen from Fig. 8 that the PCM yields a satisfactory accuracy even for such complex flow. Conclusion The flow uncertainly may be influenced by a number of factors, such as flow nonlinearity, bounded domain, the boundary conditions, and (pumping/injection) wells, recharge (e.g. rainfall or evaporation), and nonstationary trends in the aquifer parameters. However, most existing stochastic methods are insufficient to cope with such complexities. In this work, we first discussed the application of the probabilistic collocation method to the unconfined flow. Two random fields (hydraulic conductivity and recharge) are regarded as uncorrelated or fully correlated fields. We demonstrated how to select the collocation points for the uncorrelated or fully correlated cases, respectively. It is easy to extend the approach to cases with multiple random inputs. Secondly, the robustness of the probabilistic collocation method was tested with several complex examples. The numerical experiments demonstrated in this study show that the inclusion of the recharge (either deterministic or stochastic), wells (either pumping or injecting), and nonstationary trends in conductivity would lead to complex distributions of the hydraulic head moments. However, with a relatively small computational cost, the probabilistic collocation method provides a robust approach for handling nonlinear, nonstationary groundwater flow problems. In summary, the probabilistic collocation method is feasible for modeling the flow uncertainty in complex formations subject to multiple sources/sinks. The existing codes can be used directly in the probabilistic collocation method. The work presented here examined the possibility for the collocation method to be a practical tool for uncertainty qualification in the routine groundwater modeling. Acknowledgements This work is partially supported by Natural Science Foundation of China (NSFC) under Grants 40672164, 0620631, and 50688901. And the first and second author would like to acknowledge the support by China ‘‘973” Program through Grant 2006CB403406. References Bear, J., 1972. Dynamics of Fluids in Porous Media. Dover, Mineola, NY. Ding, Y., Li, T., Zhang, D., Zhang, P., 2008. Adaptive Stroud stochastic collocation method for flow in random porous media via Karhunen–Loeve expansion. Commun. Comput. Phys. 4 (1), 102–123.
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